How do you create a mesh made up of mutliple objects within each other?
If I understand correctly what you're trying to make, you can duplicate mesh with Shift+D. Then S to scale inner part to the required size and G to grab and move it.
To connect them you could select 2 edges and press F (only one edge at the time) or use Bridge Edge Loops tool to select two face loops from both meshes and bridge them with faces.
To preview what you're doing use wireframe shading mode (with Z). To have control over meshes, use Mask modifier to toggle their visibility;  Knife tool to cut half of the cup and mirror another part with Mirror modifier; store different meshes on different 3D layers among other tools.